What is the integral of e^{3x}x^2 ?

The integral of e^{3x}x^2 is 1/3 e^{3x}x^2-2/27 (3e^{3x}x-e^{3x})+C

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integral of e^{3x}x^2

\int e^{3 x} x^{2} d x

\frac{1}{3} e^{3 x} x^{2} - \frac{2}{27} (3 e^{3 x} x - e^{3 x}) + C

Solution steps

\int e^{3 x} x^{2} d x

Apply Integration By Parts

: \frac{1}{3} e^{3 x} x^{2} - \int \frac{2}{3} e^{3 x} x d x

= \frac{1}{3} e^{3 x} x^{2} - \int \frac{2}{3} e^{3 x} x d x

\int \frac{2}{3} e^{3 x} x d x = \frac{2}{27} (3 e^{3 x} x - e^{3 x})

= \frac{1}{3} e^{3 x} x^{2} - \frac{2}{27} (3 e^{3 x} x - e^{3 x})

Add a constant to the solution

= \frac{1}{3} e^{3 x} x^{2} - \frac{2}{27} (3 e^{3 x} x - e^{3 x}) + C

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\frac{d}{d x} (100 x^{\frac{3}{4}} y^{\frac{1}{4}})

sin (x), [\frac{1}{9} π, \frac{1}{8} π]

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What is the integral of e^{3x}x^2 ?
The integral of e^{3x}x^2 is 1/3 e^{3x}x^2-2/27 (3e^{3x}x-e^{3x})+C